Theorem findes 4344

Description: Finite induction with explicit substitution. The first hypothesis is the basis and the second is the induction step. Theorem Schema 22 of [Suppes] p. 136. This is an alternative for Metamath 100 proof #74. (Contributed by Raph Levien, 9-Jul-2003.)

Hypotheses

Ref	Expression
findes.1	|- [. (/) / x ]. ph
findes.2	|- ( x e. om -> ( ph -> [. suc x / x ]. ph ) )

Assertion

Ref	Expression
findes	|- ( x e. om -> ph )

Proof of Theorem findes

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 2818	. 2 |- ( z = (/) -> ( [ z / x ] ph <-> [. (/) / x ]. ph ) )
2		sbequ 1761	. 2 |- ( z = y -> ( [ z / x ] ph <-> [ y / x ] ph ) )
3		dfsbcq2 2818	. 2 |- ( z = suc y -> ( [ z / x ] ph <-> [. suc y / x ]. ph ) )
4		sbequ12r 1695	. 2 |- ( z = x -> ( [ z / x ] ph <-> ph ) )
5		findes.1	. 2 |- [. (/) / x ]. ph
6		nfv 1461	. . . 4 |- F/ x y e. om
7		nfs1v 1856	. . . . 5 |- F/ x [ y / x ] ph
8		nfsbc1v 2833	. . . . 5 |- F/ x [. suc y / x ]. ph
9	7, 8	nfim 1504	. . . 4 |- F/ x ( [ y / x ] ph -> [. suc y / x ]. ph )
10	6, 9	nfim 1504	. . 3 |- F/ x ( y e. om -> ( [ y / x ] ph -> [. suc y / x ]. ph ) )
11		eleq1 2141	. . . 4 |- ( x = y -> ( x e. om <-> y e. om ) )
12		sbequ12 1694	. . . . 5 |- ( x = y -> ( ph <-> [ y / x ] ph ) )
13		suceq 4157	. . . . . 6 |- ( x = y -> suc x = suc y )
14		dfsbcq 2817	. . . . . 6 |- ( suc x = suc y -> ( [. suc x / x ]. ph <-> [. suc y / x ]. ph ) )
15	13, 14	syl 14	. . . . 5 |- ( x = y -> ( [. suc x / x ]. ph <-> [. suc y / x ]. ph ) )
16	12, 15	imbi12d 232	. . . 4 |- ( x = y -> ( ( ph -> [. suc x / x ]. ph ) <-> ( [ y / x ] ph -> [. suc y / x ]. ph ) ) )
17	11, 16	imbi12d 232	. . 3 |- ( x = y -> ( ( x e. om -> ( ph -> [. suc x / x ]. ph ) ) <-> ( y e. om -> ( [ y / x ] ph -> [. suc y / x ]. ph ) ) ) )
18		findes.2	. . 3 |- ( x e. om -> ( ph -> [. suc x / x ]. ph ) )
19	10, 17, 18	chvar 1680	. 2 |- ( y e. om -> ( [ y / x ] ph -> [. suc y / x ]. ph ) )
20	1, 2, 3, 4, 5, 19	finds 4341	1 |- ( x e. om -> ph )

Syntax hints:   -> wi 4   <-> wb 103   = wceq 1284   e. wcel 1433   [wsb 1685   [.wsbc 2815   (/)c0 3251   suc csuc 4120   omcom 4331

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-uni 3602  df-int 3637  df-suc 4126  df-iom 4332

This theorem is referenced by: (None)